Sang-il Oum (엄상일)

Abstract

Resolving in a strong sense an old problem of Gyárfás from the 1980s on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs.

Abstract

For an $n$-element matroid $M$ given by an $n \times n$ matrix representation over a finite field $\mathbb F$ and an integer $k$, we present an $(O_{k,\mathbb F}(n^2)+O(n^ω))$-time algorithm that either finds a branch-decomposition of $M$ of width at most $k$, or confirms that the branch-width of $M$ is more than $k$, where $ω< 2.3714$ is the matrix multiplication exponent, and the $O_{k,\mathbb F}(\cdot)$-notation hides factors that depend on $k$ and $\mathbb F$ in a computable manner. All previous algorithms including Hliněný and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, and Oum [SIAM J. Discrete Math. (2021)] run in at least $Ω(n^3)$ time. Moreover, if the input matrix representation is given by a standard form, our algorithm runs in $O_{k,\mathbb F}(n^2)$-time, since $O(n^ω)$-time is only needed for finding a standard form of the input matrix. When $M$ is given by an $m \times n$ matrix, the overhead for finding a standard form is $O(mn \min(m,n)^{ω-2})$. As corollaries, we obtain faster algorithms for rank-width of directed graphs and path-width of matroids represented over a fixed finite field. Furthermore, we also present an approximation algorithm for finding branch-width that works on infinite fields provided that the input matrix is of a standard form and contains a bounded number of distinct values of entries. To suggest that our algorithm is optimal, we observe that for every field $\mathbb F$, deciding whether the branch-width of a matroid represented over $\mathbb F$ is $0$ is as hard as deciding whether a square matrix over $\mathbb F$ is singular. Under the assumption that singularity testing requires $Ω(n^ω)$-time, this implies that the overhead of $O(n^ω)$ is unavoidable. We also show strengthenings of this observation to rule out some approximations under this assumption.

Abstract

We prove that for every path $P$, the class of graphs with no induced $P$ and no induced four-cycle $C_4$ is linearly $χ$-bounded. More generally, we ask for which pairs $\{T,H\}$ where $T$ is a forest and $H$ is a complete multipartite graph, every graph $G$ with no induced $T$ and no induced $H$ has chromatic number at most $C \cdot R(α(H),ω(G)+1)$ for some constant $C$ depending only on $T$ and $H$, where $R(\cdot,\cdot)$ denotes the usual Ramsey numbers. We show that this holds in the following two instances, which strengthen the case $T=P$ and $H=C_4$ mentioned above: (1) every component of $T$ is a broom and $H$ is complete multipartite; or (2) $T$ is a forest and $H$ is complete bipartite. These two unify and substantially extend a number of previous results on linear and polynomial $χ$-boundedness for various graph classes. For case (2), we also provide a new proof (with better bounds) of a recent result of Fox, Nenadov, and Pham on the existence of an induced copy of a fixed tree in a graph satisfying certain sparsity conditions.